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This June 25, 2014, file photo, shows a group of immigrants from Honduras and El Salvador who crossed the U.S.-Mexico border illegally as they are stopped in Granjeno, Texas. (AP Photo/Eric Gay, File) more >

Professors at Yale University have roiled the immigration debate with a new study calculating there are between 16 million and 30 million illegal immigrants in the U.S. — as much as three times more than most demographers figure.

The professors’ model looked at estimates of how many people came illegally, and how many people likely left, and concluded there are a lot more people who arrived than the 11 million suggested by traditional estimates. The model says the most likely figure is double that, at about 22 million.

If true, the numbers would mean U.S. officials have done a poorer job of catching illegal immigrants than imagined, and that one out of every nine people living in the U.S. is here illegally.

“Policy debates about the amount of resources to devote to this issue, and the merits of alternative policies, including deportation, amnesty, and border control, depend critically on estimates of the number of undocumented immigrants in the U.S., which sets the scale of the issue,” said the academics, Professors Jonathan S. Feinstein and Edward H. Kaplan and postdoctoral associate Mohammad Fazel-Zarandi, all at the Yale School of Management.

They published their findings in PLOS ONE, an open access scholarly journal, and sparked fierce pushback from the demographers who study the issue and say the professors’ numbers are impossible.

“We believe these new numbers represent at most an interesting academic exercise, but are ultimately greatly off-base and thus counterproductive to the public’s very real need to understand the true scope of illegal immigration and how best to address it,” analysts at the Migration Policy Institute, who were asked to do a peer review of the study, said in their response.

The number of people in the country illegally has always been a touchy question, and is perhaps even more freighted now under President Trump.

During the 2016 campaign Mr. Trump said the numbers could be as high as 30 million — drawing protests from fact-checkers who cited the traditional demographers.

A 2005 report by analysts at Bear Stearns concluded there were 20 million illegal immigrants, far outpacing the more accepted figure at the time of about 12 million.

And President Obama’s deportation chief, former U.S. Immigration and Customs Enforcement Director Sarah Saldana, once testified to Congress that the number could be 15 million.

The government does do its own estimates, but they are sporadic and lag far behind.

The most recent study by Homeland Security was released in July 2017 and dated back to Census Bureau numbers from January 2014, or more than four-and-a-half years ago. That study put the unauthorized population at 12.1 million.

They and most other demographers use what’s known as the “residual method” for figuring the unauthorized population. The general idea is to take the total number of people who claim to be foreign-born in Census Bureau survey data, then subtract the number of people who are in the country legally. The difference is deemed to be those here without permission.

Each organization has its own variation for adjusting those numbers for census undercounts and other caveats, but they show illegal immigration peaked at about 12 million a decade ago, and now rests somewhere near 11 million.

The Yale professors, though, said there’s a major problem with the survey-based methods.

“One must locate undocumented immigrants, and once located subjects must truthfully report they were foreign born,” Mr. Kaplan told The Washington Times in an email. “Obviously undocumented immigrants do not wish to be found, nor is it in their interest to reveal their place of birth.”

He said there are tens of millions of people who fall into the category of declining to answer the surveys, and he said he and his fellow professors took a different approach.

They decided to look at the number of illegal immigrants they figured arrived over the years, either jumping the border or arriving legally but overstaying their visas. They then looked at those who were deported, went back home on their own, died or otherwise left the unauthorized population.

Running the model one million times, they came up with a conservative estimate of 16.7 million unauthorized migrants, up to 29.5 million. They figured 22.1 million was the mean.

“The results of our analysis are clear: The number of undocumented immigrants in the United States is estimated to be substantially larger than has been appreciated at least in widely accepted previous estimates,” they concluded.

Robert Warren, a demographer at the Center for Migration Studies, a New York-based think tank, said the professors’ study didn’t take into account the circular nature of migration particularly late in the last decade, when it was common for illegal immigrants from Mexico to come work in the U.S. for a short time, go back home, then return for a future work season and repeat the cycle.

“Their mechanism gets the immigrants in here, but their way of getting out is flawed. They don’t take into account of enough of this short-term migration of Mexicans,” said Mr. Warren, who spent more than three decades working at the Census Bureau and then for the former Immigration and Naturalization Service.

Mr. Warren said the professors could have done basic demographic checks to realize they were so far off a realistic count.

“The policy implications are so important that they should have done that,” he said.

The demographers at MPI called the professors’ study a “thought experiment from a team of academics who specialize in management studies.”

And Steven A. Camarota, a demographer at the D.C.-based Center for Immigration Studies, said there are data points such as birth, death and school records that can be used to check the Census-based figures.

If there were millions of uncounted illegal immigrant women of child-bearing age, there should be hundreds of thousands more births showing up in hospital records — but there aren’t.

Mr. Camarota said the numbers don’t match up exactly, and there will always be some margin of error, but it’s nothing like the massive factor of two that the professors calculated.

He also said there have been real-world tests, including the 1986 amnesty which, despite massive fraud, generally produced the same legalization levels as would have been predicted from Census bureau data.

“None of that means that 11 or 12 million estimate is rock solid. It could be off. Maybe there are an extra million, maybe there are 2 million. There’s a margin of error around that number. But nearly 17 million?” he said.

He added: “When they came up with this number they should have stepped back and said we’ve got a problem, it doesn’t pass the kind of prima facie approach of what seems possible.”